106 Prof. W. E. Story on Partial 



4. For x r = = 1, 2, 3, ... , tc- 1) we have, by (29) 

 and (24), 



H I+ 4::)«'"-'"- 





^ *«r = (k = lf 2 , 3, . . .,*-! and * £ r}, 



so that, by (16), 



^^=0 (* = 1,2 8 3,...,«-1), 



and, therefore, 



xdu r = 0; (36) 



also, for x K = we have, by (29), (4), and (26), 



— . = I — ± + a K ~~ \ e = — e [k = 1, J, ,5, ...,*;— 1), 



so that, by (16), 



*jb = (* = 1,2,3,...,k-1) 



and, therefore, 



*.<&*. = ( 37 ) 



It is not self-evident that x r du r = for x r — 0, because 

 one or more of the derivatives of u might be infinite, but 



(36) and (37) here proved show that even if ^— - is infinite 



Qoc k 



for x r — 0, it is infinite of so low an order that its product 

 by oc r is 0. 



If, now, x s = 1 (5 = 1, 2, 3, . . . . , k) all the other a?'s are 

 0, by (3), and, therefore, by (36) and (37), equation (30) 

 reduces to 



du = (38) 



